The Flattened Aggregate Constraint Homotopy Method for Nonlinear ...

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Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 430932, 14 pages http://dx.doi.org/10.1155/2014/430932

Research Article The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints Zhengyong Zhou1 and Bo Yu2 1 2

School of Mathematics and Computer Sciences, Shanxi Normal University, Linfen, Shanxi 041004, China School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China

Correspondence should be addressed to Zhengyong Zhou; [email protected] Received 8 February 2014; Accepted 27 April 2014; Published 29 May 2014 Academic Editor: Victor Kovtunenko Copyright Β© 2014 Z. Zhou and B. Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The aggregate constraint homotopy method uses a single smoothing constraint instead of π‘š-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.

1. Introduction In this paper, we consider the following nonlinear programming problem: min

𝑓 (π‘₯) ,

s.t. 𝑔 (π‘₯) ≀ 0,

(1)

where π‘₯ ∈ 𝑅𝑛 is the variable, 𝑔(π‘₯) = (𝑔1 (π‘₯), . . . , π‘”π‘š (π‘₯))𝑇 , 𝑓 : 𝑅𝑛 β†’ 𝑅, and 𝑔𝑖 (π‘₯) : 𝑅𝑛 β†’ 𝑅, 𝑖 = 1, . . . , π‘š, are three times continuously differentiable, and π‘š is very large, but 𝑛 is moderate. It has wide applications, and a typical situation is the discretized semi-infinite programming problem. From the mid-1980s, much attention has been paid to interior point methods for mathematical programming, and many results on theory, algorithms, and applications on linear programming, convex programming, complementarity problems, semidefinite programming, and linear cone programming were obtained (see monographs [1–6] and

references therein). For nonlinear programming, the typical algorithms used were the Newton-type methods to the perturbed first-order necessary conditions combined with line search or trust region methods with a proper merit function (e.g., [7–9]). The general conditions of global convergence for these methods required that the feasible set be bounded and that the Jacobian matrix be uniformly nonsingular. Another typical class of globally convergent methods for nonlinear programming was probability-one homotopy methods (e.g., [10–12]), whose global convergence can be established under some weaker conditions than the ones for Newton-type methods. The excellent feature is that, unlike line search or trust region methods, they do not depend on the descent of a merit function and so are insensitive to the local minimum of the merit function, in which any search direction is not a descent direction of the merit function. In [10, 11], Feng et al. proposed a homotopy method for nonlinear programming (1), which was called the combined homotopy interior point (abbreviated by CHIP) method;

2

Abstract and Applied Analysis

its global convergence was proven under the normal cone condition (see below for its definition) for the feasible set as well as some common conditions. On the basis of the CHIP method, some modified CHIP methods were presented in [13, 14]; the global convergence was established under the quasinormal cone and pseudocone condition for the feasible set, respectively. In [12], Watson described some probabilityone homotopy methods for the unconstrained and inequality constrained optimization, whose global convergence was established under some weaker assumptions. Recently, Yu and Shang proposed a constraint shifting combined homotopy method in [15, 16], in which not only the objective function but also the constraint functions were regularly deformed. The global convergence was proven under the condition that the initial feasible set, which approaches the feasible set of (1) as the homotopy parameter changes from 1 to 0, not necessarily the feasible set of (1), satisfies the normal cone condition. Let 𝑔max (π‘₯) = max1β‰€π‘–β‰€π‘š {𝑔𝑖 (π‘₯)}; then (1) is equivalent to min s.t.

𝑓 (π‘₯) , 𝑔max (π‘₯) ≀ 0,

(2)

which has only one, but nonsmooth, constraint. In [17], the following aggregate function was introduced, which is a smooth approximation of 𝑔max (π‘₯) with a smoothing parameter 𝑑 > 0 and induced from the max-entropy theory: π‘š

𝑔̂ (π‘₯, 𝑑) = 𝑑 ln βˆ‘ exp ( 𝑖=1

𝑔𝑖 (π‘₯) ). 𝑑

(3)

It is also known as exponential penalty function (see [18]). By using it for all constraint functions of problem (1), an aggregate constraint homotopy (abbreviated by ACH) method was presented by Yu et al. in [19], whose global convergence was obtained under the condition that the feasible set satisfies the weak normal cone condition. Although the ACH method has only one smoothing constraint, the gradient and Hessian of the aggregate constraint function π‘š

βˆ‡π‘₯ 𝑔̂ (π‘₯, 𝑑) = βˆ‘π‘π‘– (π‘₯, 𝑑) βˆ‡π‘”π‘– (π‘₯) , 𝑖=1

π‘š πœ•π‘”Μ‚ (π‘₯, 𝑑) 1 = (𝑔̂ (π‘₯, 𝑑) βˆ’ βˆ‘π‘π‘– (π‘₯, 𝑑) 𝑔𝑖 (π‘₯)) , πœ•π‘‘ 𝑑 𝑖=1 2 𝑔̂ (π‘₯, 𝑑) βˆ‡π‘₯π‘₯ π‘š

= βˆ‘π‘π‘– (π‘₯, 𝑑) βˆ‡2 𝑔𝑖 (π‘₯) 𝑖=1

1π‘š 𝑇 + βˆ‘π‘π‘– (π‘₯, 𝑑) βˆ‡π‘”π‘– (π‘₯) (βˆ‡π‘”π‘– (π‘₯)) 𝑑 𝑖=1 π‘š 1π‘š 𝑇 βˆ’ βˆ‘π‘π‘– (π‘₯, 𝑑) βˆ‡π‘”π‘– (π‘₯) βˆ‘π‘π‘– (π‘₯, 𝑑) (βˆ‡π‘”π‘– (π‘₯)) , 𝑑 𝑖=1 𝑖=1

2 𝑔̂ (π‘₯, 𝑑) βˆ‡π‘₯𝑑

=

π‘š 1 π‘š ( 𝑐 𝑐𝑖 (π‘₯, 𝑑) 𝑔𝑖 (π‘₯) 𝑑) βˆ‡π‘” βˆ‘ βˆ‘ (π‘₯, (π‘₯) 𝑖 𝑑2 𝑖=1 𝑖 𝑖=1 π‘š

βˆ’βˆ‘π‘π‘– (π‘₯, 𝑑) 𝑔𝑖 (π‘₯) βˆ‡π‘”π‘– (π‘₯)) , 𝑖=1

(4) where 𝑐𝑖 (π‘₯, 𝑑) =

exp (𝑔𝑖 (π‘₯) /𝑑) ∈ (0, 1) , exp (𝑔𝑖 (π‘₯) /𝑑)

βˆ‘π‘š 𝑖=1

π‘š

βˆ‘π‘π‘– (π‘₯, 𝑑) = 1, 𝑖=1

(5) are complicated combinations of gradients and Hessians of all constraint functions and, hence, are expensive to calculate when π‘š is very large. Throughout this paper, we assume that the nonlinear programming problem (1) possesses a very large number of nonlinear constraints, but a small number of variables, and the objective and constraint functions are not sparse. For such a problem, the number of constraint functions can be so large that the computation of gradients and Hessians of all constraint functions is very expensive and cannot be stored in memory and, hence, the general numerical methods for solving nonlinear programming are not efficient. Although active set methods only need to calculate gradients and Hessians of a part of constraint functions, require lower storage, and have faster numerical solution, the working set is difficult to estimate without knowing the internal structure of the problem. In this paper, we present a new homotopy method called the flattened aggregate constraint homotopy (abbreviated by FACH) method for nonlinear programming (1) by using a new smoothing technique, in which only a part of constraint functions is aggregated. Under the normal cone condition for the feasible set and some other general assumptions, we prove that the FACH method can determine a smooth homotopy path from a given interior point of the feasible set to a KKT point of (1), and preliminary numerical results demonstrate its efficiency. The rest of this paper is organized as follows. We conclude this section with some notations, definitions, and a lemma. The flattened aggregate constraint function with some properties is given in Section 2. The homotopy map, existence and convergence of a smooth homotopy path with proof are given in Section 3. A numerical procedure for tracking the smooth homotopy path, and numerical test results with some remarks are given in Section 4. Finally, we conclude the paper with some remarks in Section 5. When discussing scalars and scalar-valued functions, subscripts refer to iteration step so that superscripts can be used for exponentiation. In contrast, for vectors and vectorvalued functions, subscripts are used to indicate components, whereas superscripts are used to indicate the iteration step. The identity matrix is represented by 𝐼. Unless otherwise specified, β€– β‹… β€– denotes the Euclidean norm. Ξ© = {π‘₯ ∈ 𝑅𝑛 | 𝑔(π‘₯) ≀ 0} is the feasible set of (1),

Abstract and Applied Analysis

3

whereas Ξ©0 = {π‘₯ ∈ 𝑅𝑛 | 𝑔(π‘₯) < 0} is the interior of Ξ© and πœ•Ξ© = Ξ© \ Ξ©0 is the boundary of Ξ©. The symbols 𝑅+π‘š π‘š and 𝑅++ denote the nonnegative and positive quadrants of π‘š 𝑅 , respectively. The active index set is denoted by 𝐼(π‘₯) = {𝑖 ∈ {1, . . . , π‘š} | 𝑔𝑖 (π‘₯) = 0} at π‘₯ ∈ Ξ©. For a function 𝐹 : 𝑅𝑛 β†’ π‘…π‘š , βˆ‡πΉ(π‘₯) is the 𝑛 Γ— π‘š matrix whose (𝑖, 𝑗)th element is πœ•πΉπ‘— (π‘₯)/πœ•π‘₯𝑖 ; πΉβˆ’1 (π‘Œ) = {π‘₯ | 𝐹(π‘₯) ∈ π‘Œ} is the setvalued inverse for the set π‘Œ βŠ† π‘…π‘š . Definition 1 (see [10]). The set Ξ© satisfies the normal cone condition if and only if the normal cone of Ξ© at any π‘₯ ∈ πœ•Ξ© does not meet Ξ©0 ; that is, 0

{π‘₯ + βˆ‘ 𝑦𝑖 βˆ‡π‘”π‘– (π‘₯) | 𝑦𝑖 β‰₯ 0} ∩ Ξ© = 0.

(6)

constraints, which is weaker than the linear independence constraint qualification. The last assumption, the normal cone condition of the feasible set, is a generalization of the convex condition for the feasible set. Indeed, if Ξ© is a convex set, then it satisfies the normal cone condition. Some simple nonconvex sets, satisfying the normal cone condition, are shown in [10]. In this paper, we construct a flattened aggregate constraint function as follows: 𝑔̂(πœƒ,𝑐1 ,𝑐2 ,𝛼) (π‘₯, 𝑑) π‘š

= πœƒπ‘‘ ln (βˆ‘πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp ( 𝑖=1

π‘–βˆˆπΌ(π‘₯)

Definition 2. If and only if there exists πœ†βˆ— ∈ 𝑅+π‘š such that (π‘₯βˆ— , πœ†βˆ— ) satisfies βˆ‡π‘“ (π‘₯βˆ— ) + βˆ‡π‘” (π‘₯βˆ— ) πœ†βˆ— = 0, πœ†βˆ—π‘– 𝑔𝑖

βˆ—

πœ†βˆ—π‘–

(π‘₯ ) = 0,

βˆ—

β‰₯ 0, 𝑔𝑖 (π‘₯ ) ≀ 0, 𝑖 = 1, . . . , π‘š,

Definition 3. Let π‘ˆ βŠ‚ 𝑅𝑛 be an open set and let 𝐹 : π‘ˆ β†’ π‘…π‘š be a differentiable map. We say 𝑦 ∈ π‘…π‘š is a regular value of 𝐹 if and only if Range [

πœ•πΉ (π‘₯) ] = π‘…π‘š , πœ•π‘₯

βˆ€π‘₯ ∈ πΉβˆ’1 (𝑦) .

where πœ€ (𝑑) = 𝑐1 𝑑 + 𝑐2 ,

(7)

then π‘₯βˆ— is called a KKT point of (1) and πœ†βˆ— is the corresponding Lagrangian multiplier.

(8)

Lemma 4 (parameterized Sard theorem [20]). Let π‘ˆ βŠ‚ 𝑅𝑛 and 𝑉 βŠ‚ π‘…π‘˜ be two open sets and let 𝐹 : π‘ˆ Γ— 𝑉 β†’ π‘…π‘š be a πΆπ‘Ÿ differentiable map with π‘Ÿ > max{0, 𝑛 βˆ’ π‘š}. If 0 ∈ π‘…π‘š is a regular value of 𝐹, then, for almost all π‘Ž ∈ 𝑉, 0 is a regular value of πΉπ‘Ž = 𝐹(β‹…, π‘Ž).

0, for 𝑧 ≀ βˆ’π›Όπœ€ (𝑑) with 𝛼 > 1, { { πœ‘ (𝑧, 𝑑) = {1, for 𝑧 β‰₯ βˆ’πœ€ (𝑑) , { πœ‘ 𝑑) , otherwise, (𝑧, {

πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0,

πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) = 1,

Assumption 5. Ξ©0 is nonempty and Ξ© is bounded.

πœ•2 πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) = 0, πœ•π‘§2

βˆ‘ 𝑦𝑖 βˆ‡π‘”π‘– (π‘₯) = 0,

π‘–βˆˆπΌ(π‘₯)

𝑦𝑖 β‰₯ 0, 𝑖 ∈ 𝐼 (π‘₯) 󳨐⇒ 𝑦𝑖 = 0, 𝑖 ∈ 𝐼 (π‘₯) . (9)

Assumption 7. Ξ© satisfies the normal cone condition (see Definition 1). The first assumption is the Slater’s condition and the boundedness of the feasible set, which are two basic conditions. The second assumption provides the regularity of

(12)

where, for any 𝑑 ∈ [0, 1], πœ‘(𝑧, 𝑑) satisfies

In this section, we suppose that the following assumptions hold.

Assumption 6. For any π‘₯ ∈ πœ•Ξ©, {βˆ‡π‘”π‘– (π‘₯) | 𝑖 ∈ 𝐼(π‘₯)} are positive independent; that is,

(11)

𝑐1 > 0, 𝑐2 > 0, 0 < πœƒ ≀ 1 are some adjusting parameters, πœ‘(𝑧, 𝑑) : 𝑅×[0, 1] β†’ [0, 1] is a 𝐢3 bivariate function satisfying πœ‘(𝑧, 𝑑) = 0 for 𝑧 ≀ βˆ’π›Όπœ€(𝑑) with 𝛼 > 1, and πœ‘(𝑧, 𝑑) = 1 for 𝑧 β‰₯ βˆ’πœ€(𝑑). For simplicity of discussion, we write 𝑔̂(πœƒ,𝑐1 ,𝑐2 ,𝛼) (π‘₯, 𝑑) Μ‚ 𝑑). By its definition, πœ‘(𝑧, 𝑑) can be expressed as as 𝑔(π‘₯,

πœ•2 πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0, πœ•π‘§2

2. The Flattened Aggregate Constraint Function

𝑔𝑖 (π‘₯) πœ€ (𝑑) ) + exp (βˆ’ )) , πœƒπ‘‘ πœƒπ‘‘ (10)

πœ•πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0, πœ•π‘§ πœ•3 πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0, πœ•π‘§3 πœ•πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) = 0, πœ•π‘§

(13)

πœ•3 πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) = 0. πœ•π‘§3

There exist many kinds of functions πœ‘(𝑧, 𝑑), such as polynomial functions and spline functions. Since polynomial functions have simple mathematical expressions and many excellent properties, throughout this paper, πœ‘(𝑧, 𝑑) is chosen as πœ‘ (𝑧, 𝑑) =

βˆ’20(𝑧 + πœ€ (𝑑))7 βˆ’70(𝑧 + πœ€ (𝑑))6 + (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))7 (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))6 βˆ’84(𝑧 + πœ€ (𝑑))5 βˆ’35(𝑧 + πœ€ (𝑑))4 + + + 1. (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))5 (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))4

(14)

4

Abstract and Applied Analysis

Let πΌπœ€ (π‘₯, 𝑑) = {𝑖 | 𝑔𝑖 (π‘₯) > βˆ’π›Όπœ€(𝑑)}; by the definition of Μ‚ 𝑑) in (10) can be rewritten as πœ‘(𝑧, 𝑑), 𝑔(π‘₯,

1 β‰₯ βˆ‘ πœ† 𝑖 (π‘₯, 𝑑)

𝑔 (π‘₯) ) 𝑔̂ (π‘₯, 𝑑) = πœƒπ‘‘ ln ( βˆ‘ πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp ( 𝑖 πœƒπ‘‘ π‘–βˆˆπΌ (π‘₯,𝑑) πœ€

+ exp (βˆ’

(2) By limπ‘₯β†’π‘₯,𝑑→𝑑 (βˆ’πœ€(𝑑) βˆ’ π‘”π‘–βˆ— (π‘₯)) = βˆ’πœ€(𝑑) < 0, we have

π‘–βˆˆπΌπœ€ (π‘₯,𝑑)

(15) πœ€ (𝑑) )). πœƒπ‘‘

=

π‘–βˆˆπΌπœ€ (π‘₯,𝑑)

(16)

βˆ‘π‘—βˆˆπΌπœ€ (π‘₯,𝑑) πœ‘ (𝑔𝑗 (π‘₯) , 𝑑) exp (𝑔𝑗 (π‘₯) /πœƒπ‘‘) + exp (βˆ’πœ€ (𝑑) /πœƒπ‘‘)

β‰₯

πœ‘ (π‘”π‘–βˆ— (π‘₯) , 𝑑) exp (π‘”π‘–βˆ— (π‘₯) /πœƒπ‘‘) πœ‘ (π‘”π‘–βˆ— (π‘₯) , 𝑑) exp (π‘”π‘–βˆ— (π‘₯) /πœƒπ‘‘) + exp (βˆ’πœ€ (𝑑) /πœƒπ‘‘)

β‰₯

πœ‘ (π‘”π‘–βˆ— (π‘₯) , 𝑑) 1 + exp ((βˆ’πœ€ (𝑑) βˆ’ π‘”π‘–βˆ— (π‘₯)) /πœƒπ‘‘)

Μ‚ 𝑑) with respect to π‘₯ is The gradient of 𝑔(π‘₯, βˆ‡π‘₯ 𝑔̂ (π‘₯, 𝑑) = βˆ‘ (πœ† 𝑖 (π‘₯, 𝑑) + πœ‰π‘– (π‘₯, 𝑑)) βˆ‡π‘”π‘– (π‘₯) ,

βˆ‘π‘–βˆˆπΌπœ€ (π‘₯,𝑑) πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp (𝑔𝑖 (π‘₯) /πœƒπ‘‘)

󳨀→ 1. (20)

where πœ† 𝑖 (π‘₯, 𝑑) =

πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp (𝑔𝑖 (π‘₯) /πœƒπ‘‘)

, βˆ‘π‘—βˆˆπΌπœ€ (π‘₯,𝑑) πœ‘ (𝑔𝑗 (π‘₯) , 𝑑) exp (𝑔𝑗 (π‘₯) /πœƒπ‘‘) + exp (βˆ’πœ€ (𝑑) /πœƒπ‘‘) (17)

πœ‰π‘– (π‘₯, 𝑑) =

Hence, βˆ‘π‘–βˆˆπΌπœ€ (π‘₯,𝑑) πœ† 𝑖 (π‘₯, 𝑑) β†’ 1. Together with item 1, we have βˆ‘π‘–βˆˆπΌ(π‘₯) πœ† 𝑖 (π‘₯, 𝑑) β†’ 1 for πœƒπ‘‘ ↓ 0, 𝑑 β†’ 𝑑, and π‘₯ β†’ π‘₯. (3) If 𝑔𝑖 (π‘₯) β‰₯ βˆ’πœ€(𝑑), then πœ•πœ‘(𝑔𝑖 (π‘₯), 𝑑)/πœ•π‘§ = 0 by its definition, and hence πœ‰π‘– (π‘₯, 𝑑) = 0;

(21)

else

πœƒπ‘‘πœ•πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) /πœ•π‘§ exp (𝑔𝑖 (π‘₯) /πœƒπ‘‘)

. βˆ‘π‘—βˆˆπΌπœ€ (π‘₯,𝑑) πœ‘ (𝑔𝑗 (π‘₯) , 𝑑) exp (𝑔𝑗 (π‘₯) /πœƒπ‘‘) + exp (βˆ’πœ€ (𝑑) /πœƒπ‘‘) (18)

󡄨 󡄨 󡄨󡄨 󡄨 πœƒπ‘‘ σ΅„¨σ΅„¨πœ•πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) /πœ•π‘§σ΅„¨σ΅„¨σ΅„¨ exp (𝑔𝑖 (π‘₯) /πœƒπ‘‘) ≀ πœƒπ‘‘π‘€, σ΅„¨σ΅„¨πœ‰π‘– (π‘₯, 𝑑)󡄨󡄨󡄨 ≀ 󡄨 exp (βˆ’πœ€ (𝑑) /πœƒπ‘‘) (22)

Μ‚ 𝑑) only relate to a part Then the gradient and Hessian of 𝑔(π‘₯, of constraint functions; that is, 𝑖 ∈ πΌπœ€ (π‘₯, 𝑑). Proposition 8. For any πœƒ > 0, 𝑑 ∈ (0, 1], π‘₯ ∈ Ξ© with πœƒπ‘‘ ↓ 0, 𝑑 β†’ 𝑑 ∈ [0, 1], π‘₯ β†’ π‘₯ ∈ πœ•Ξ©,

where 𝑀 = max(𝑧,𝑑)βˆˆπ‘…Γ—[0,1] |πœ•πœ‘(𝑧, 𝑑)/πœ•π‘§|. Then, we have πœ‰(π‘₯, 𝑑) β†’ 0 for πœƒπ‘‘ ↓ 0, 𝑑 β†’ 𝑑, and π‘₯ β†’ π‘₯. Proposition 9. For any given 0 < πœƒ ≀ 1,

(1) πœ† 𝑖 (π‘₯, 𝑑) β†’ 0 for 𝑖 βˆ‰ 𝐼(π‘₯);

Μ‚ 𝑑) ≀ max{𝑔max (π‘₯), βˆ’πœ€(𝑑)} + πœƒπ‘‘ ln(π‘šπΌπœ€ + 1), (1) βˆ’πœ€(𝑑) ≀ 𝑔(π‘₯,

(2) βˆ‘π‘–βˆˆπΌ(π‘₯) πœ† 𝑖 (π‘₯, 𝑑) β†’ 1;

Μ‚ 𝑑) ≀ 𝑔max (π‘₯) + πœƒπ‘‘ ln(π‘šπΌπœ€ + 1) (2) 𝑔max (π‘₯) ≀ 𝑔(π‘₯, when 𝑔max (π‘₯) β‰₯ βˆ’πœ€(𝑑),

(3) πœ‰(π‘₯, 𝑑) β†’ 0. Proof. (1) Because π‘₯ ∈ πœ•Ξ©, there exists π‘–βˆ— ∈ 𝐼(π‘₯) such that π‘”π‘–βˆ— (π‘₯) = 0. By the continuity of πœ‘(𝑧, 𝑑) and 𝑔(π‘₯), we know that limπ‘₯β†’π‘₯,𝑑→𝑑 πœ‘(π‘”π‘–βˆ— (π‘₯), 𝑑) = πœ‘(π‘”π‘–βˆ— (π‘₯), 𝑑) = 1 and limπ‘₯β†’π‘₯ (𝑔𝑖 (π‘₯) βˆ’ π‘”π‘–βˆ— (π‘₯)) = 𝑔𝑖 (π‘₯) < 0 for 𝑖 βˆ‰ 𝐼(π‘₯); hence

Μ‚ 𝑑) = βˆ’πœ€(𝑑) when 𝑔max (π‘₯) ≀ βˆ’π›Όπœ€(𝑑) with 𝛼 > 1, (3) 𝑔(π‘₯, where π‘šπΌπœ€ denotes the cardinality of πΌπœ€ (π‘₯, 𝑑). Proof. (1) By πœ‘(𝑧, 𝑑) β‰₯ 0, we have

0 ≀ πœ† 𝑖 (π‘₯, 𝑑) exp (𝑔𝑖 (π‘₯) /πœƒπ‘‘) ≀ πœ‘ (π‘”π‘–βˆ— (π‘₯) , 𝑑) exp (π‘”π‘–βˆ— (π‘₯) /πœƒπ‘‘) exp ((𝑔𝑖 (π‘₯) βˆ’ π‘”π‘–βˆ— (π‘₯)) /πœƒπ‘‘) = πœ‘ (π‘”π‘–βˆ— (π‘₯) , 𝑑) 󳨀→ 0. Hence, item 1 is satisfied for πœƒπ‘‘ ↓ 0, 𝑑 β†’ 𝑑, and π‘₯ β†’ π‘₯.

𝑔̂ (π‘₯, 𝑑) = πœƒπ‘‘ ln ( βˆ‘ πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp ( π‘–βˆˆπΌπœ€ (π‘₯,𝑑)

(19)

+ exp (βˆ’ β‰₯ πœƒπ‘‘ ln (exp (βˆ’ = βˆ’πœ€ (𝑑) .

πœ€ (𝑑) )) πœƒπ‘‘

πœ€ (𝑑) )) πœƒπ‘‘

𝑔𝑖 (π‘₯) ) πœƒπ‘‘

(23)

Abstract and Applied Analysis

5

Then the left inequality can be obtained. By πœ‘(𝑧, 𝑑) ≀ 1, we have 𝑔̂ (π‘₯, 𝑑) = πœƒπ‘‘ ln ( βˆ‘ πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp ( π‘–βˆˆπΌπœ€ (π‘₯,𝑑)

πœƒ ∈ (0, 1] satisfying πœƒ ≀ βˆ’ max{𝑔max (π‘₯βˆ— ), βˆ’πœ€(1)}/(2 ln(π‘š+1)), Μ‚ by Proposition 9(1), for any π‘₯ ∈ Ξ©, 𝑔̂ (π‘₯, 1) ≀ max {𝑔max (π‘₯) , βˆ’πœ€ (1)} + πœƒ ln (π‘š + 1)

𝑔𝑖 (π‘₯) ) πœƒπ‘‘

≀ max {𝑔max (π‘₯βˆ— ) , βˆ’πœ€ (1)} + πœƒ ln (π‘š + 1)

πœ€ (𝑑) + exp (βˆ’ )) πœƒπ‘‘

=

1 max {𝑔max (π‘₯βˆ— ) , βˆ’πœ€ (1)} 2

(27)

< 0, 𝑔 (π‘₯) πœ€ (𝑑) ≀ πœƒπ‘‘ ln ( βˆ‘ exp ( 𝑖 ) + exp (βˆ’ )) . πœƒπ‘‘ πœƒπ‘‘ π‘–βˆˆπΌ (π‘₯,𝑑) πœ€

(24)

Μ‚ βŠ‚ Ξ©πœƒ (1)0 . which means Ξ© (3) For any π‘₯ ∈ Ξ©, that is, 𝑔max (π‘₯) ≀ 0, by the right inequality of Proposition 9(1), lim 𝑔̂ (π‘₯, 𝑑) ≀ lim max {𝑔max (π‘₯) , βˆ’πœ€ (𝑑)}

Then the right inequality can be obtained by Proposition 2.3 in [19]. (2) For any π‘₯, 𝑑 with 𝑔max (π‘₯) β‰₯ βˆ’πœ€(𝑑), let π‘”π‘–βˆ— (π‘₯) = 𝑔max (π‘₯); then πœ‘(π‘”π‘–βˆ— (π‘₯), 𝑑) = 1 by its definition, and hence

𝑑↓0

𝑑↓0

(28)

≀ max {𝑔max (π‘₯) , βˆ’π‘2 } ≀ 0, where the second inequality comes from βˆ’πœ€(𝑑) = βˆ’π‘1 𝑑 βˆ’ 𝑐2 ≀ βˆ’π‘2 < 0 for 𝑑 ∈ [0, 1], which means Ξ© βŠ‚ lim𝑑↓0 Ξ©πœƒ (𝑑). Then, together with item 1,

𝑔 (π‘₯) 𝑔̂ (π‘₯, 𝑑) = πœƒπ‘‘ ln ( βˆ‘ πœ‘ (𝑔𝑖 (π‘₯) , 𝑑) exp ( 𝑖 ) πœƒπ‘‘ π‘–βˆˆπΌ (π‘₯,𝑑) πœ€

+ exp (βˆ’

β‰₯ πœƒπ‘‘ ln (πœ‘ (π‘”π‘–βˆ— (π‘₯) , 𝑑) exp ( = πœƒπ‘‘ ln (exp (

lim Ξ©πœƒ (𝑑) = Ξ©.

πœ€ (𝑑) )) πœƒπ‘‘ π‘”π‘–βˆ— (π‘₯) )) πœƒπ‘‘

𝑑↓0

(25)

π‘”π‘–βˆ— (π‘₯) )) πœƒπ‘‘

Proposition 11. There exists a πœƒ ∈ (0, 1] such that, for any Μ‚ 𝑑) =ΜΈ 0. 𝑑 ∈ (0, 1], π‘₯ ∈ πœ•Ξ©πœƒ (𝑑); then βˆ‡π‘₯ 𝑔(π‘₯, Proof. If not, for any πœƒ ∈ (0, 1], there exist corresponding Μ‚ πœƒ , π‘‘πœƒ ) = 0, which π‘‘πœƒ ∈ (0, 1] and π‘₯πœƒ ∈ πœ•Ξ©πœƒ (π‘‘πœƒ ) such that βˆ‡π‘₯ 𝑔(π‘₯ means that there must exist three sequences {πœƒπ‘˜ }∞ π‘˜=1 βŠ‚ (0, 1], π‘˜ ∞ π‘˜ {π‘‘π‘˜ }∞ βŠ‚ (0, 1], and {π‘₯ } with π‘₯ ∈ πœ•Ξ© (𝑑 πœƒπ‘˜ π‘˜ ), such that π‘˜=1 π‘˜=1 π‘˜ πœƒπ‘˜ ↓ 0, π‘‘π‘˜ β†’ 𝑑, π‘₯ β†’ π‘₯ ∈ πœ•Ξ©, and

= π‘”π‘–βˆ— (π‘₯) = 𝑔max (π‘₯) ; then the left inequality is true. The right inequality can be obtained by item 1. Μ‚ 𝑑). (3) It is trivial by the definition of 𝑔(π‘₯, Μ‚ 𝑑) ≀ 0} and Ξ©πœƒ (𝑑)0 = Proposition 10. Let Ξ©πœƒ (𝑑) = {π‘₯ | 𝑔(π‘₯, Μ‚ 𝑑) < 0}; one has {π‘₯ | 𝑔(π‘₯, (1) Ξ©πœƒ (𝑑) βŠ‚ Ξ© for 𝑑 ∈ (0, 1]; Μ‚ βŠ‚ Ξ©0 , there exists (2) for any bounded and closed set Ξ© Μ‚ a πœƒ ∈ (0, 1], such that Ξ© βŠ‚ Ξ©πœƒ (1)0 ; (3) Ξ©πœƒ (𝑑) β†’ Ξ© as 𝑑 ↓ 0. Proof. (1) If π‘₯ βˆ‰ Ξ©, which is equivalent to 𝑔max (π‘₯) > 0, by Proposition 9(2), 0 < 𝑔max (π‘₯) ≀ 𝑔̂ (π‘₯, 𝑑) ,

(29)

(26)

which means π‘₯ βˆ‰ Ξ©πœƒ (𝑑); then we have Ξ©πœƒ (𝑑) βŠ‚ Ξ©. Μ‚ is a (2) By the continuity of 𝑔(π‘₯) and the fact that Ξ© βˆ— Μ‚ bounded closed set, there exists a point π‘₯ ∈ Ξ© such that Μ‚ and 𝑔max (π‘₯βˆ— ) < 0. For any 𝑔max (π‘₯) reaches its maximum in Ξ©,

βˆ‡π‘₯ 𝑔̂ (π‘₯π‘˜ , π‘‘π‘˜ ) =

βˆ‘ (πœ† 𝑖 (π‘₯π‘˜ , π‘‘π‘˜ ) + πœ‰π‘– (π‘₯π‘˜ , π‘‘π‘˜ )) βˆ‡π‘”π‘– (π‘₯π‘˜ ) 󳨀→ 0 π‘–βˆˆπΌπœ€ (π‘₯π‘˜ ,π‘‘π‘˜ )

(30)

as π‘˜ β†’ ∞. Then, by Proposition 8, we know that πœ†(π‘₯π‘˜ , π‘‘π‘˜ ) β†’ πœ†, πœ‰(π‘₯π‘˜ , π‘‘π‘˜ ) β†’ 0 as π‘˜ β†’ ∞, and βˆ‘

πœ†π‘– = βˆ‘ πœ†π‘– = 1,

π‘–βˆˆlimπ‘˜σ³¨€β†’βˆž πΌπœ€ (π‘₯π‘˜ ,π‘‘π‘˜ )

π‘–βˆˆπΌ(π‘₯)

(31)

where the first equality comes from that πœ†π‘– = 0 for 𝑖 βˆ‰ 𝐼(π‘₯). Therefore, we have βˆ‘π‘–βˆˆπΌ(π‘₯) πœ†π‘– βˆ‡π‘”π‘– (π‘₯) = 0 by taking limits on (30); this is a contradiction to Assumption 6. Μ‚ βŠ‚ Ξ©0 , there Proposition 12. For any bounded closed set Ξ© exists a πœƒ ∈ (0, 1] such that Μ‚ = 0, {π‘₯ + πœ†βˆ‡π‘₯ 𝑔̂ (π‘₯, 1) | πœ† > 0} ∩ Ξ© for any π‘₯ ∈ πœ•Ξ©πœƒ (1).

(32)

6

Abstract and Applied Analysis

Μ‚ and four Proof. If not, there exist a bounded closed set Ξ© ∞ π‘˜ ∞ π‘˜ sequences {πœƒπ‘˜ }π‘˜=1 βŠ‚ (0, 1], {π‘₯ }π‘˜=1 with π‘₯ ∈ πœ•Ξ©πœƒπ‘˜ (1), π‘˜ ∞ π‘˜ Μ‚ {π‘₯Μ‚π‘˜ }∞ π‘˜=1 βŠ‚ Ξ©, and {πœ† }π‘˜=1 > 0 such that πœƒπ‘˜ ↓ 0, π‘₯ β†’ π‘₯ ∈ πœ•Ξ©, Μ‚ πœ† 𝑖 (π‘₯π‘˜ , 1) β†’ πœ†βˆ— β‰₯ 0, πœ‰π‘– (π‘₯π‘˜ , 1) β†’ 0 as π‘˜ β†’ ∞, π‘₯Μ‚π‘˜ β†’ π‘₯Μ‚ ∈ Ξ©, 𝑖 and π‘₯Μ‚π‘˜ = π‘₯π‘˜ + πœ†π‘˜

βˆ‘ (πœ† 𝑖 (π‘₯π‘˜ , 1) + πœ‰π‘– (π‘₯π‘˜ , 1)) βˆ‡π‘”π‘– (π‘₯π‘˜ ) . π‘–βˆˆπΌπœ€ (π‘₯π‘˜ ,1) (33)

𝐻 (π‘₯, πœ†, 𝑑) (1 βˆ’ 𝑑) (βˆ‡π‘“ (π‘₯) + πœ†βˆ‡π‘₯ 𝑔̂ (π‘₯, 𝑑)) + 𝑑 (π‘₯ βˆ’ π‘₯0 ) ) =( πœ†π‘”Μ‚ (π‘₯, 𝑑) βˆ’ π‘‘πœ†0 𝑔̂ (π‘₯0 , 1)

(39)

= 0,

By Proposition 8, we have βˆ‘

3.1. The Flattened Aggregate Constraint Homotopy. Using Μ‚ 𝑑) in (10), the flattened aggregate constraint function 𝑔(π‘₯, we construct the following flattened aggregate constraint homotopy:

πœ†βˆ—π‘– = βˆ‘ πœ†βˆ—π‘– = 1,

π‘–βˆˆlimπ‘˜σ³¨€β†’βˆž πΌπœ€ (π‘₯π‘˜ ,1)

π‘–βˆˆπΌ(π‘₯)

(34)

where the first equality comes from that πœ†βˆ—π‘– = 0 for 𝑖 βˆ‰ 𝐼(π‘₯). Then, we have βˆ‘ (πœ† 𝑖 (π‘₯π‘˜ , 1) + πœ‰π‘– (π‘₯π‘˜ , 1)) βˆ‡π‘”π‘– (π‘₯π‘˜ ) π‘–βˆˆπΌπœ€ (π‘₯π‘˜ ,1)

(35)

󳨀→ βˆ‘ πœ†βˆ—π‘– βˆ‡π‘”π‘– (π‘₯) . π‘–βˆˆπΌ(π‘₯)

By Assumption 6 and using (34), βˆ‘ πœ†βˆ—π‘– βˆ‡π‘”π‘– (π‘₯) =ΜΈ 0.

π‘–βˆˆπΌ(π‘₯)

(36)

Hence {πœ†π‘˜ }∞ π‘˜=1 is a bounded sequence, and there must exist a subsequence converging to πœ† > 0; then we have by (33) π‘₯Μ‚ = π‘₯ + πœ† βˆ‘ πœ†βˆ—π‘– βˆ‡π‘”π‘– (π‘₯) , π‘–βˆˆπΌ(π‘₯)

(37)

which contradicts Assumption 7.

3. The Flattened Aggregate Constraint Homotopy Method In [19], the following aggregate constraint homotopy was introduced: 𝐻 (π‘₯, πœ†, 𝑑) (1 βˆ’ 𝑑) (βˆ‡π‘“ (π‘₯) + πœ†βˆ‡π‘₯ π‘”Μ‚πœƒ (π‘₯, 𝑑)) + 𝑑 (π‘₯ βˆ’ π‘₯0 ) ) (38) =( πœ†π‘”Μ‚πœƒ (π‘₯, 𝑑) βˆ’ π‘‘πœ†0 π‘”Μ‚πœƒ (π‘₯0 , 1) = 0, where π‘”Μ‚πœƒ (π‘₯, 𝑑) = πœƒπ‘‘ ln βˆ‘π‘š 𝑖=1 exp(𝑔𝑖 (π‘₯)/πœƒπ‘‘) with πœƒ ∈ (0, 1], πœ† is the Lagrangian multiplier of the aggregate constraint Μ‚ Γ— 𝑅++ . function π‘”Μ‚πœƒ (π‘₯, 𝑑), and (π‘₯0 , πœ†0 ) is the starting point in Ξ© Under the weak normal cone condition, which is similar to the normal cone condition, it was proved that the ACH determines a smooth interior path from a given interior point to a KKT point. Then, the predictor-corrector procedure can be applied to trace the homotopy path from (π‘₯0 , πœ†0 , 1) to (π‘₯βˆ— , πœ†βˆ— , 0), in which (π‘₯βˆ— , πœ†βˆ— ) is a KKT point of (1).

where πœ† is the Lagrangian multiplier of the flattened aggregate constraint and (π‘₯0 , πœ†0 ) is the starting point and can be Μ‚ Γ— 𝑅++ . randomly chosen from Ξ© We give the main theorem on the existence and convergence of a smooth path from (π‘₯0 , πœ†0 , 1) to (π‘₯βˆ— , πœ†βˆ— , 0), in which (π‘₯βˆ— , πœ†βˆ— ) is a solution of the KKT system of (1), and hence the global convergence of our proposal, namely, the FACH method, can be proven. Theorem 13. Suppose that Assumptions 5–7 hold and πœƒ ∈ Μ‚βŠ‚ (0, 1] satisfies Propositions 10–12 for the bounded closed set Ξ© 0 0 0 0 0 Ξ©πœƒ (1) βŠ‚ Ξ© ; then for almost all starting points 𝑀 = (π‘₯ , πœ† ) ∈ Μ‚ Γ— 𝑅++ , the zero-points set π»βˆ’1 (0) = {(π‘₯, πœ†, 𝑑) | 𝐻(π‘₯, πœ†, 𝑑) = Ξ© 0} defines a smooth path Γ𝑀0 , which starts at (π‘₯0 , πœ†0 , 1) and approaches the hyperplane 𝑑 = 0. Furthermore, let (π‘₯βˆ— , πœ†βˆ— , 0) be any limit point of Γ𝑀0 , and π‘¦βˆ— = πœ†βˆ— πœ†(π‘₯βˆ— , 0) (πœ†(π‘₯βˆ— , 0) is a limit of πœ†(π‘₯, 𝑑) as π‘₯ β†’ π‘₯βˆ— , 𝑑 ↓ 0), then π‘₯βˆ— is a KKT point of (1) and π‘¦βˆ— is the corresponding Lagrangian multiplier. Proof. Consider 𝐻 as a map of the variable (𝑀0 , π‘₯, πœ†, 𝑑), for Μ‚ Γ— 𝑅++ Γ— (0, 1], which means π‘₯0 ∈ Ξ©πœƒ (1)0 ; any (𝑀0 , 𝑑) ∈ Ξ© 0 Μ‚ , 1) < 0, because hence 𝑔(π‘₯ πœ•π» (𝑀0 , π‘₯, πœ†, 𝑑) πœ•π‘€0

βˆ’π‘‘πΌ 0 𝑇 =( 0 ) 0 βˆ’π‘‘πœ† (βˆ‡π‘₯ 𝑔̂ (π‘₯ , 1)) βˆ’π‘‘π‘”Μ‚ (π‘₯0 , 1) (40)

is nonsingular and the Jacobi matrix of 𝐻(𝑀0 , π‘₯, πœ†, 𝑑) is of full row rank. Using the parameterized Sard theorem, Lemma 4, Μ‚ Γ— 𝑅++ , 0 is a regular value of 𝐻(π‘₯, πœ†, 𝑑). for almost all 𝑀0 ∈ Ξ© From π‘₯ βˆ’ π‘₯0 ), 𝐻 (π‘₯, πœ†, 1) = ( πœ†π‘”Μ‚ (π‘₯, 1) βˆ’ πœ†0 𝑔̂ (π‘₯0 , 1)

(41)

we know that (π‘₯0 , πœ†0 ) is the unique and simple solution of 𝐻(π‘₯, πœ†, 1) = 0. Then, by the implicit function theorem, there exists a smooth curve Γ𝑀0 βŠ‚ Ξ©πœƒ (𝑑)0 ×𝑅++ Γ—(0, 1] starting from (π‘₯0 , πœ†0 , 1) and being transversal to the hyperplane 𝑑 = 1. Since Γ𝑀0 can be extended in Ξ©πœƒ (𝑑)0 Γ— 𝑅++ Γ— (0, 1) until it converges to the boundary of Ξ©πœƒ (𝑑)0 Γ— 𝑅++ Γ— (0, 1), there must exist an extreme point (π‘₯, πœ†, 𝑑) ∈ πœ•(Ξ©πœƒ (𝑑) Γ— 𝑅+ Γ— [0, 1]). Let (π‘₯, πœ†, 𝑑) be any extreme point of Γ𝑀0 other than (π‘₯0 , πœ†0 , 1); then only the following five cases are possible: (1) (π‘₯, πœ†, 𝑑) ∈ Ξ© Γ— 𝑅+ Γ— {0}, πœ† < +∞;

Abstract and Applied Analysis

7 Let (π‘₯βˆ— , πœ†βˆ— , πœ†(π‘₯βˆ— , 0)) be any accumulation point; π‘¦π‘–βˆ— = πœ†βˆ— πœ† 𝑖 (π‘₯βˆ— , 0). By (39) and the fact that πœ‰(π‘₯, 𝑑) β†’ 0 as 𝑑 ↓ 0, we have

(2) (π‘₯, πœ†, 𝑑) ∈ Ξ©πœƒ (1) Γ— 𝑅+ Γ— {1}, πœ† < +∞; (3) (π‘₯, πœ†, 𝑑) ∈ πœ•Ξ©πœƒ (𝑑) Γ— 𝑅++ Γ— (0, 1), πœ† < +∞; (4) (π‘₯, πœ†, 𝑑) ∈ Ξ©πœƒ (𝑑) Γ— {0} Γ— (0, 1);

π‘š

βˆ‡π‘“ (π‘₯βˆ— ) + βˆ‘π‘¦π‘–βˆ— βˆ‡π‘”π‘– (π‘₯βˆ— ) = 0.

(5) (π‘₯, πœ†, 𝑑) ∈ Ξ©πœƒ (𝑑) Γ— {+∞} Γ— [0, 1]. Case (2) means that 𝐻(π‘₯, πœ†, 1) = 0 has another solution (π‘₯, πœ†) except (π‘₯0 , πœ†0 ), or (π‘₯0 , πœ†0 ) is a double solution of 𝐻(π‘₯, πœ†, 1) = 0, which contradicts the fact that (π‘₯0 , πœ†0 ) is the unique and simple solution of 𝐻(π‘₯, πœ†, 1) = 0. Case (3) means Μ‚ 𝑑) = 0, 0 < πœ† < ∞, and 0 < 𝑑 < 1; hence 𝑔(π‘₯, πœ†π‘”Μ‚ (π‘₯, 𝑑) βˆ’ π‘‘πœ†0 𝑔̂ (π‘₯0 , 1) = βˆ’π‘‘πœ†0 𝑔̂ (π‘₯0 , 1) < 0,

(42)

which contradicts the last equation of (39). Case (4) means Μ‚ 𝑑) ≀ 0, πœ† = 0, and 0 < 𝑑 < 1; hence 𝑔(π‘₯, πœ†π‘”Μ‚ (π‘₯, 𝑑) βˆ’ π‘‘πœ†0 𝑔̂ (π‘₯0 , 1) = βˆ’π‘‘πœ†0 𝑔̂ (π‘₯0 , 1) < 0,

(43)

which contradicts the last equation of (39). Then, Cases (2), (3), and (4) are impossible. Because 𝐻(π‘₯, πœ†, 1) = 0 has only one and also unique solution (π‘₯0 , πœ†0 ), Case (2) is impossible. By the continuity and the last equation of (39), we know that Cases (3) and (4) are impossible. If Case (5) holds, there must exist a sequence π‘˜ β†’ π‘₯, πœ†π‘˜ β†’ ∞, {(π‘₯π‘˜ , πœ†π‘˜ , π‘‘π‘˜ )}∞ π‘˜=1 on Γ𝑀0 such that π‘₯ and π‘‘π‘˜ β†’ 𝑑. By the last equation of (39), we have 𝑔̂ (π‘₯π‘˜ , π‘‘π‘˜ ) =

π‘‘π‘˜ πœ†0 𝑔̂ (π‘₯0 , 1) πœ†π‘˜

󳨀→ 0

(44)

as π‘˜ β†’ ∞, which means π‘₯ ∈ πœ•Ξ©πœƒ (𝑑). The following two cases may be possible. (1) If 𝑑 = 1, let 𝛼 be any accumulation point of {(1 βˆ’ π‘‘π‘˜ )πœ†π‘˜ }, which is known to exist from (39) and Μ‚ π‘˜ , π‘‘π‘˜ ) =ΜΈ 0; then we have limπ‘˜β†’βˆž βˆ‡π‘₯ 𝑔(π‘₯ π‘₯0 = π‘₯ + π›Όβˆ‡π‘₯ 𝑔̂ (π‘₯, 1) .

(45)

If 𝛼 = 0, then π‘₯ = π‘₯0 contradicts π‘₯0 ∈ Ξ©πœƒ (1)0 ; else contradicts Proposition 12.

(47)

𝑖=1

By the fact that lim𝑑↓0 Ξ©πœƒ (𝑑) = Ξ©, we have π‘₯βˆ— ∈ Ξ©. If π‘₯βˆ— ∈ Ξ© , we know that 0

lim 𝑔̂ (π‘₯, 𝑑)

π‘₯󳨀→π‘₯βˆ— ,𝑑↓0

≀

lim

π‘₯󳨀→π‘₯βˆ— ,𝑑↓0

(max {𝑔max (π‘₯) , βˆ’πœ€ (𝑑)} + πœƒπ‘‘ ln (π‘šπΌπœ€ + 1))

= max {𝑔max (π‘₯βˆ— ) , πœ€ (0)} = max {𝑔max (π‘₯βˆ— ) , βˆ’π›Όπ‘2 } < 0, (48) where the first inequality comes from Proposition 9(1), the second equality comes from the continuity, and the third equality comes from the definition of πœ€(𝑑) in (11); hence πœ†βˆ— = 0 Μ‚ 𝑑) = by the last equation of (39); else π‘₯βˆ— ∈ πœ•Ξ©, limπ‘₯β†’π‘₯βˆ— ,𝑑↓0 𝑔(π‘₯, 𝑔max (π‘₯βˆ— ) = 0 by Proposition 9(2) and πœ† 𝑖 (π‘₯βˆ— , 0) = 0 for 𝑖 βˆ‰ 𝐼(π‘₯βˆ— ) by Proposition 8. Thus we have π‘¦π‘–βˆ— 𝑔𝑖 (π‘₯βˆ— ) = 0,

𝑖 = 1, . . . , π‘š.

(49)

Summing up, (π‘₯βˆ— , π‘¦βˆ— ) is a solution of the KKT system of (1), which means that π‘₯βˆ— is a KKT point of (1) and π‘¦βˆ— is the corresponding Lagrangian multiplier. 3.2. The Modified Flattened Aggregate Constraint Homotopy. From Proposition 8(3), we know that πœ‰(π‘₯, 𝑑) β†’ 0 as 𝑑 ↓ 0 for any π‘₯ ∈ Ξ© and, hence, we can use the following modified flattened aggregate constraint homotopy (MFACH) instead of the FACH: (1 βˆ’ 𝑑) (βˆ‡π‘“ (π‘₯) + πœ†β„Ž (π‘₯, 𝑑)) + 𝑑 (π‘₯ βˆ’ π‘₯0 ) ) 𝐻 (π‘₯, πœ†, 𝑑) = ( πœ†π‘”Μ‚ (π‘₯, 𝑑) βˆ’ π‘‘πœ†0 𝑔̂ (π‘₯0 , 1) = 0,

(2) If 𝑑 < 1, by (39), we have

(50)

(1 βˆ’ 𝑑) lim πœ†π‘˜ βˆ‡π‘₯ 𝑔̂ (π‘₯π‘˜ , π‘‘π‘˜ ) = βˆ’ (1 βˆ’ 𝑑) βˆ‡π‘“ (π‘₯) βˆ’ 𝑑 (π‘₯ βˆ’ π‘₯0 ) ,

where

π‘˜σ³¨€β†’βˆž

(46) the right-hand is finite; however, by π‘˜ Μ‚ , π‘‘π‘˜ ) =ΜΈ 0, we know that the left-hand is limπ‘˜β†’βˆž βˆ‡π‘₯ 𝑔(π‘₯ infinite; this is a contradiction. As a conclusion, Case (1) is the only possible case. This implies that Γ𝑀0 must approach the hyperplane 𝑑 = 0. π‘˜ ∞ π‘˜ ∞ Because {πœ†(π‘₯π‘˜ , π‘‘π‘˜ )}∞ π‘˜=1 (defined in (17)), {π‘₯ }π‘˜=1 , and {πœ† }π‘˜=1 are bounded sequences, we know that {(π‘₯π‘˜ , πœ†π‘˜ , πœ†(π‘₯π‘˜ , π‘‘π‘˜ ))}∞ π‘˜=1 has at least one accumulation point as π‘˜ β†’ ∞.

β„Ž (π‘₯, 𝑑) = βˆ‘ πœ† 𝑖 (π‘₯, 𝑑) βˆ‡π‘”π‘– (π‘₯) . π‘–βˆˆπΌπœ€ (π‘₯,𝑑)

(51)

Remarks. (i) Since πœ‰(π‘₯, 𝑑) is dropped in (51), the expressions of the homotopy map (50) and its Jacobian and hence the corresponding code become simpler. Moreover, the computation of H in (50) and its Jacobi matrix DH is a little cheaper than that for the FACH method. (ii) To guarantee that 𝐻(π‘₯, πœ†, 𝑑) in (39) be a 𝐢2 map, πœ‘(𝑧, 𝑑) must be 𝐢3 ; hence, if πœ‘(𝑧, 𝑑) is chosen as a polynomial, the

8

Abstract and Applied Analysis

total degree should be seven; in contrast, for the homotopy map in (50), πœ‘(𝑧, 𝑑) is only needed to be 𝐢2 . Then πœ‘(𝑧, 𝑑) should satisfy πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0,

πœ•πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0, πœ•π‘§

πœ•2 πœ‘ (βˆ’π›Όπœ€ (𝑑) , 𝑑) = 0, πœ•π‘§2 πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) = 1,

πœ•πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) = 0, πœ•π‘§

(52)

βˆ‡π‘“ (π‘₯) + πœ†β„Ž (π‘₯, 𝑑𝑐 ) ) = 0, 𝐹𝑑𝑐 (π‘₯, πœ†) = ( πœ†π‘”Μ‚ (π‘₯, 𝑑𝑐 )

πœ•2 πœ‘ (βˆ’πœ€ (𝑑) , 𝑑) =0 πœ•π‘§2

6(𝑧 + πœ€ (𝑑))5 15(𝑧 + πœ€ (𝑑))4 + (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))5 (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))4 10(𝑧 + πœ€ (𝑑))3 + + 1. (π›Όπœ€ (𝑑) βˆ’ πœ€ (𝑑))3

(54)

where 𝑑𝑐 is a small positive constant. For other situations, the steplength will be decreased to make new predictor-corrector steps (see Algorithm 1).

and πœ‘(𝑧, 𝑑) can be chosen as πœ‘ (𝑧, 𝑑) =

the predictor direction. The corrector step terminates when 𝑑(π‘˜,𝑖) and 𝐻(𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ) satisfy the tolerance criterions. At each predictor step and corrector step, the feasibility of the predictor point (π‘€π‘˜ , π‘‘π‘˜ ) and the corrector point (𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ) needs to be checked. If π‘‘π‘˜ < 𝑑𝑐 in the predictor step or π‘‘π‘˜ < 0 in the corrector step, a damping step is used to get a new point (𝑀(π‘˜,0) , 0). Then, if 𝑀(π‘˜,0) is feasible, the end game, a more efficient strategy than predictor-corrector steps when the homotopy parameter is close to 0, is invoked. Starting with 𝑀(π‘˜,0) , Newton’s method is used to solve

(53)

(iii) The existence and convergence of the homotopy path defined by the MFACH can be proven in a similar way with that of the FACH.

4. The FACH-S-N Procedure and Numerical Results 4.1. The FACH-S-N Procedure. In this section, we give a numerical procedure, FACH-S-N procedure, to trace the flattened aggregate constraint homotopy path by secant predictor and Newton corrector steps. It consists of three main steps: the predictor step, the corrector step, and the end game. The predictor step is an approximate step along the homotopy path: it uses a predictor direction π‘‘π‘˜ and a steplength β„Žπ‘˜ to get a predictor point. The first predictor direction uses the tangent direction, and others use the secant direction. The steplength is determined by several parameters. It is set to no more than 1, which can ensure that the predictor point is close to the homotopy path and hence stays in the convergence domain of Newton’s method in the corrector step. If the angle of the predictor direction and the previous one π›½π‘˜ is greater than πœ‹/4, the steplength will be decreased to avoid using the opposite direction as the predictor direction. If the corrector criteria are satisfied with no more than four Newton iterations for three times in succession, the steplength will be increased or kept invariable. Otherwise, the steplength will be decreased. Once a predictor point (𝑀(π‘˜,0) , 𝑑(π‘˜,0) ) is calculated, one or more Newton iterations are used to bring the predictor point back to the homotopy path in the corrector step. The corrector points (𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ), 𝑖 = 1, . . . , 5, are calculated by (𝑀(π‘˜,𝑖+1) , 𝑑(π‘˜,𝑖+1) ) = (𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ) + 𝑑(π‘˜,𝑖+1) , 𝑖 = 0, . . . , 4, where the step 𝑑(π‘˜,𝑖+1) is the solution of an augmented system defined by the homotopy equation and the direction perpendicular to

4.2. The Numerical Experiment. Although there exist so many test problems, such as the CUTEr test set [21], we cannot find a large collection of test problems with moderate variables and very many complicated nonlinear constraints. In this paper, six test problems are chosen to test the algorithm. Problem 4.1 is chosen from the CUTEr test set and it is used to illustrate a special situation; others are derived from the discretized semi-infinite programming problems. We also give two artificial test problems 4.2 and 4.3 and use three problems 4.4–4.6 in [22]. The numbers of variables in problem 4.2 and the number of constrains in problems 4.2– 4.6 can be arbitrary. For each test problem, the gradients and Hessians of the objective and constraint functions are evaluated analytically. The FACH-S-N procedure was implemented in MATLAB. To illustrate its efficiency, we also implemented the ACH method in [19] using a similar procedure. In addition, we downloaded the state-of-the-art solver KNITRO, which provides three algorithms for solving large-scale nonlinear programming problems, and we used the interior-point direct method with default parameters to compare with the FACH and MFACH methods. For any iterate point π‘₯π‘˜ , if 𝑔max (π‘₯π‘˜ ) < 10βˆ’6 , it was treated as a feasible point. The test results were obtained by running MATLAB R2008a on a desktop with Windows XP Professional operation system, Intel(R) Core(TM) i5-750 2.66 GHz processor, and 8 GB of memory. The default parameters were chosen as follows. (i) Parameters for the flattened aggregate constraint function: πœƒ = 0.01, 𝑐1 = 0.05, 𝑐2 = 0.5 βˆ— 10βˆ’5 , and 𝛼 = 2. (ii) Parameters in the end game section: 𝑑𝑐 = 10βˆ’6 and 𝑑end = 0.1. (iii) Step size parameters: β„Ž0 = 0.1, 𝐡min = [0.5, 0.75], and 𝐡max = [3, 1.5]. (iv) Tracking tolerances: 𝐻tol = 10βˆ’5 and 𝐻final = 10βˆ’12 . (v) Initial Lagrangian multipliers: 1 for ACH, FACH, and MFACH methods.

Abstract and Applied Analysis

9

Μ‚ Γ— 𝑅++ ; initial steplength β„Ž1 , steplength Input Give πœƒ ∈ (0, 1], 𝑐1 > 0, 𝑐2 > 0, 𝛼 > 1, 𝑑end and 𝑑𝑐 ; starting point 𝑀0 ∈ Ξ© contraction factor 𝐡min , steplength expansion factor 𝐡max ; tracking tolerances 𝐻tol and 𝐻final for correction. Set 𝑑0 = 1, π‘˜ = 1, IT = 0, 𝑁good = 2, 𝛽1 = 0. Step 1. The predictor step. Step 1.1. Compute the predictor direction. If π‘˜ = 1, compute 𝑑 by solving 𝐻󸀠 (π‘₯0 , πœ†0 , 1) ) 𝑑 = βˆ’π‘‘0 , ( 𝑇 (𝑑0 ) where 𝑑0 = (0, . . . , 0, βˆ’1) ∈ 𝑅𝑛+2 , set the predictor direction 𝑑1 = 𝑑/ ‖𝑑‖; else, compute the predictor direction π‘‘π‘˜ , the angle π›½π‘˜ between π‘‘π‘˜ and π‘‘π‘˜βˆ’1 : 𝑇 π‘˜βˆ’1 (π‘€π‘˜βˆ’1 , π‘‘π‘˜βˆ’1 ) βˆ’ (π‘€π‘˜βˆ’2 , π‘‘π‘˜βˆ’2 ) π‘‘π‘˜ = , π›½π‘˜ = arccos ((π‘‘π‘˜ ) 𝑑 ) . π‘˜βˆ’1 π‘˜βˆ’2 β€–(𝑀 , π‘‘π‘˜βˆ’1 ) βˆ’ (𝑀 , π‘‘π‘˜βˆ’2 )β€– Step 1.2. Adjust the steplength. Adjust the steplength β„Žπ‘˜ as follows: if the predictor point is infeasible, set β„Žπ‘˜ = 𝐡min (1)β„Žπ‘˜ , 𝑁good = 0; else if the corrector step fails or π›½π‘˜ > πœ‹/4, set β„Žπ‘˜ = 𝐡min (1)β„Žπ‘˜βˆ’1 , 𝑁good = 0; else if 𝑖 = 5, set β„Žπ‘˜ = 𝐡min (2)β„Žπ‘˜βˆ’1 , 𝑁good = 0; else if 𝑖 = 4, set β„Žπ‘˜ = β„Žπ‘˜βˆ’1 , 𝑁good = 𝑁good + 1; else if 𝑖 = 3, set 𝑁good = 𝑁good + 1, if 𝑁good > 2, set β„Žπ‘˜ = min{1, 𝐡max (2)β„Žπ‘˜βˆ’1 }, else, set β„Žπ‘˜ = β„Žπ‘˜βˆ’1 ; else, set 𝑁good = 𝑁good + 1, if 𝑁good > 2, set β„Žπ‘˜ = min{1, 𝐡max (1)β„Žπ‘˜βˆ’1 }, else, set β„Žπ‘˜ = β„Žπ‘˜βˆ’1 . If β„Žπ‘˜ < 10βˆ’10 , stop the algorithm with an error flag. Step 1.3. Compute the predictor point and check its feasibility. Compute the predictor point (𝑀(π‘˜,0) , 𝑑(π‘˜,0) ) = (π‘€π‘˜βˆ’1 , π‘‘π‘˜βˆ’1 ) + β„Žπ‘˜ π‘‘π‘˜ . 0 (π‘˜,0) βˆ‰ Ξ©πœƒ (𝑑(π‘˜,0) ) Γ— 𝑅++ or 𝑑(π‘˜,0) β‰₯ 1, If 𝑀 the predictor point is infeasible, goto Step 1.2; else if 𝑑(π‘˜,0) ≀ 𝑑end , adjust the steplength β„Žπ‘˜ , compute the point (𝑀(π‘˜,0) , 0), if 𝑀(π‘˜,0) ∈ Ξ© Γ— 𝑅+ , goto Step 3, else, the predictor point is infeasible, goto Step 1.2; else, goto Step 2. Step 2. The corrector step. Set 𝑖 = 0. Repeat If 𝑖 = 5, set IT = IT + 5, π‘€π‘˜ = π‘€π‘˜βˆ’1 , π‘‘π‘˜ = π‘‘π‘˜βˆ’1 , π‘‘π‘˜+1 = π‘‘π‘˜ , π‘˜ = π‘˜ + 1, the corrector step fails, goto Step 1.2. Compute the Newton step 𝑑(π‘˜,𝑖+1) by solving 𝐻󸀠 (𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ) (π‘˜,𝑖+1) βˆ’π» (𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ) )𝑑 =( ), ( π‘˜ 𝑇 0 (𝑑 ) the corrector point (𝑀(π‘˜,𝑖+1) , 𝑑(π‘˜,𝑖+1) ) = (𝑀(π‘˜,𝑖) , 𝑑(π‘˜,𝑖) ) + 𝑑(π‘˜,𝑖+1) ; set 𝑖 = 𝑖 + 1. If 𝑀(π‘˜,𝑖) βˆ‰ Ξ©πœƒ (𝑑(π‘˜,𝑖) )0 Γ— 𝑅++ or 𝑑(π‘˜,𝑖) β‰₯ 1, set IT = IT + 𝑖, π‘€π‘˜ = π‘€π‘˜βˆ’1 , π‘‘π‘˜ = π‘‘π‘˜βˆ’1 , π‘‘π‘˜+1 = π‘‘π‘˜ , π‘˜ = π‘˜ + 1, the corrector step fails, goto Step 1.2; else if 𝑑(π‘˜,𝑖) < 0, compute the point (𝑀(π‘˜,0) , 0) by a damping Newton step, set IT = IT + 𝑖, if 𝑀(π‘˜,0) ∈ Ξ© Γ— 𝑅+ , goto Step 3, else, set π‘€π‘˜ = π‘€π‘˜βˆ’1 , π‘‘π‘˜ = π‘‘π‘˜βˆ’1 , π‘‘π‘˜+1 = π‘‘π‘˜ , π‘˜ = π‘˜ + 1, the corrector step fails, goto Step 1.2. (π‘˜,𝑖) Until ‖𝐻(𝑀 , 𝑑(π‘˜,𝑖) )β€–inf ≀ 𝐻tol and ‖𝑑(π‘˜,𝑖) β€– ≀ 𝐻tol . If π‘‘π‘˜ < 𝑑𝑐 , return with π‘₯βˆ— = π‘₯π‘˜ , stop the algorithm; else, set 𝐻tol = min{𝐻tol , π‘‘π‘˜ }, π‘˜ = π‘˜ + 1, goto Step 1. Step 3. The end game. Set 𝑙 = 0. Repeat If 𝑙 = 5 or ‖𝑑(π‘˜,𝑙) β€– > ‖𝑑(π‘˜,π‘™βˆ’1) β€– for 𝑙 > 1, or 𝑀(π‘˜,𝑙) βˆ‰ Ξ© Γ— 𝑅+ , set 𝑑end = 0.3 βˆ— 𝑑end , π‘€π‘˜ = π‘€π‘˜βˆ’1 , π‘‘π‘˜ = π‘‘π‘˜βˆ’1 , π‘‘π‘˜+1 = π‘‘π‘˜ , IT = IT + 𝑙, π‘˜ = π‘˜ + 1, the corrector step fails, goto Step 1.2. Compute the Newton step 𝑑(π‘˜,𝑙+1) by solving 𝐹𝑑󸀠𝑐 (𝑀(π‘˜,𝑙) )𝑑(π‘˜,𝑙+1) = βˆ’πΉπ‘‘π‘ (𝑀(π‘˜,𝑙) ), the corrector point 𝑀(π‘˜,𝑙+1) = 𝑀(π‘˜,𝑙) + 𝑑(π‘˜,𝑙+1) ; Algorithm 1: Continued.

10

Abstract and Applied Analysis

set 𝑙 = 𝑙 + 1. Until ‖𝐹𝑑𝑐 (𝑀(π‘˜,𝑙) )β€–inf ≀ 𝐻final and ‖𝑑(π‘˜,𝑙) β€– ≀ 𝐻final . Set IT = IT + 𝑙, return with π‘₯βˆ— = π‘₯(π‘˜,𝑙) , stop the algorithm. Algorithm 1: The FACH-S-N procedure.

For each problem with different parameters, we list the value of objective function 𝑓(π‘₯) and the maximal function of constraints 𝑔max (π‘₯) at π‘₯βˆ— , the number of iterations 𝐼𝑇, the number of evaluations of gradients of individual constraint functions in the FACH and MFACH methods 𝑁𝑠 (in contrast, 𝑁𝑠 is π‘š βˆ— 𝐼𝑇 in the ACH method), and the running time in seconds π‘‡π‘–π‘šπ‘’. For problems that were not solved by the conservative setting, we also give the reason for failure. The notation β€œfail1 ” indicates that the steplength in predictor step is smaller than 10βˆ’10 ; it is generally due to poor conditioned Jacobian matrix. The notation β€œfail2 ” means out of memory. The notation β€œfail3 ” means no result in 5000 iterations or 3600 seconds.

Table 1: Test results for Example 14. Method ACH KNITRO FACH MFACH

2 ), βˆ‘ 100 sin (βˆ’π‘₯𝑖 + 1.5πœ‹ + π‘₯π‘–βˆ’1

(55)

+

(𝑑𝑗󸀠 βˆ’ π‘₯2 ) π‘₯42

βˆ’ 1,

𝑑𝑖 =

𝑖 , (βˆšπ‘š βˆ’ 1)

𝑖 = 0, . . . , βˆšπ‘š βˆ’ 1,

𝑑𝑗󸀠 =

𝑗 , (βˆšπ‘š βˆ’ 1)

𝑗 = 0, . . . , βˆšπ‘š βˆ’ 1,

2≀𝑖≀1000

(57)

π‘₯0 = (0, 0, 100, 100) ∈ 𝑅4 . Example 17. Consider

π‘₯0 = (1, . . . , 1) ∈ 𝑅1000 . Remark 15. In this example, the starting point happens to be an unconstrained minimizer of the objective function and we found that the π‘₯-components of iterative points generated by the FACH and MFACH methods (not other homotopy methods) remain invariant. This is not an occasional phenomenon. In fact, if (π‘₯0 , 0) ∈ Ξ©0 Γ— 𝑅+π‘š is a solution of the KKT system, that is, π‘₯0 is a stationary point of the objective function 𝑓(π‘₯), when the parameters 𝛼, 𝑐1 , and 𝑐2 in FACH and MFACH methods satisfy 𝑔max (π‘₯0 ) ≀ βˆ’π›Ό(𝑐1 + 𝑐2 ), the π‘₯-components of points on the homotopy path Γ𝑀0 will remain invariant, which can be derived from the fact that (π‘₯0 , π‘‘πœ†0 πœ€(1)/πœ€(𝑑), 𝑑) is the solution of (39) and (50) for 𝑑 ∈ (0, 1]. Moreover, since 𝐻󸀠 (π‘₯0 , πœ†0 , 1) = (

Time 85.5496 2.8733 8.3912 8.4150

2

2

(𝑑 βˆ’ π‘₯ ) 𝑔𝑖,𝑗 (π‘₯) = 𝑖 2 1 π‘₯3

𝑖 = 1; 𝑖 = 2, . . . , 1000; 𝑖 = 1001; 𝑖 = 1002, . . . , 2000,

𝑁𝑠 β€” β€” 0 0

IT 35 0 4 4

𝑓 (π‘₯) = π‘₯32 + π‘₯42 ,

𝑓 (π‘₯) = sin (π‘₯1 βˆ’ 1 + 1.5πœ‹)

π‘₯1 βˆ’ πœ‹ { { 2 { {π‘₯π‘–βˆ’1 βˆ’ π‘₯𝑖 βˆ’ πœ‹ 𝑔𝑖 (π‘₯) = { { βˆ’π‘₯ βˆ’ πœ‹ { { 12 {βˆ’π‘₯π‘–βˆ’1 + π‘₯𝑖 βˆ’ πœ‹

𝑔max (π‘₯βˆ— ) βˆ’2.1416 βˆ’2.1416 βˆ’2.1416 βˆ’2.1416

Example 16. Consider

Example 14 (see [21]). Consider

+

𝑓(π‘₯βˆ— ) βˆ’99901.0000 βˆ’99901.0000 βˆ’99901.0000 βˆ’99901.0000

𝐼 0 0 ), 0 𝑔̂ (π‘₯0 , 1) βˆ—

(56)

the π‘₯-component of 𝑑1 is 0. By a similar discussion, the π‘₯component of 𝑑(1,𝑖) is 0, and hence π‘₯1 = π‘₯(1,𝑖) = π‘₯0 . In this analogy, we know that π‘₯π‘˜ = π‘₯(π‘˜,𝑖) = π‘₯0 for any π‘˜ and 𝑖 in algorithm FACH-S-N.

𝑓 (π‘₯) =

1 2 βˆ‘ (π‘₯ βˆ’ 1) , 𝑛 1β‰€π‘˜β‰€π‘› π‘˜

𝑔𝑖 (π‘₯) = ∏ cos (𝑑𝑖 π‘₯π‘˜ ) + 𝑑𝑖 βˆ‘ π‘₯π‘˜3 , 𝑑𝑖 =

1β‰€π‘˜β‰€π‘›

1β‰€π‘˜β‰€π‘›

0.5 + πœ‹π‘– , (π‘š βˆ’ 1)

𝑖 = 0, . . . , π‘š βˆ’ 1,

(58)

π‘₯0 = (βˆ’2, . . . , βˆ’2) ∈ 𝑅𝑛 . Example 18 (see [22]). Consider 𝑓 (π‘₯) = π‘₯12 + π‘₯22 + π‘₯32 , 𝑔𝑖 (π‘₯) = π‘₯1 + π‘₯2 exp (π‘₯3 𝑑𝑖 ) + exp (2𝑑𝑖 ) βˆ’ 2 sin (4𝑑𝑖 ) , 𝑑𝑖 =

𝑖 , (π‘š βˆ’ 1)

𝑖 = 0, . . . , π‘š βˆ’ 1,

π‘₯0 = (βˆ’200, βˆ’200, 200) ∈ 𝑅3 .

(59)

Abstract and Applied Analysis

11 Table 2: Test results for Example 16.

π‘š

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

102

104

106

βˆ—

𝑓(π‘₯ ) 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

𝑔max (π‘₯βˆ— ) βˆ’1.3863e βˆ’ 08 βˆ’7.1233e βˆ’ 10 βˆ’1.3863e βˆ’ 08 βˆ’1.3863e βˆ’ 08 βˆ’1.3863e βˆ’ 08 βˆ’8.0005e βˆ’ 10 βˆ’1.3863e βˆ’ 08 βˆ’1.3863e βˆ’ 08 βˆ’1.3863e βˆ’ 08 βˆ’7.8830e βˆ’ 10 βˆ’1.3863e βˆ’ 08 βˆ’1.3863e βˆ’ 08

IT 394 18 541 567 397 20 541 567 404 13 541 567

𝑁𝑠 β€” β€” 721 823 β€” β€” 721 823 β€” β€” 805 907

Time 0.2423 0.2288 0.3270 0.2692 1.9730 0.7227 0.4359 0.4676 249.3395 2170.4301 10.9486 11.2815

Table 3: (a) Test results for Example 17 with 𝑛 = 100. (b) Test results for Example 17 with 𝑛 = 500. (c) Test results for Example 17 with 𝑛 = 1000. (d) Test results for Example 17 with 𝑛 = 2000. (a)

π‘š 102

103

104

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

βˆ—

𝑓(π‘₯ ) 1.4918 0.2603 1.4915 1.4915 1.4918 0.2603 1.4915 1.4915 1.4918 0.2603 1.4915 1.4915

𝑔max (π‘₯βˆ— ) βˆ’1.5282e βˆ’ 03 βˆ’1.9708e βˆ’ 06 βˆ’6.6613e βˆ’ 16 6.1062e βˆ’ 15 βˆ’1.5282e βˆ’ 03 βˆ’2.7042e βˆ’ 05 βˆ’6.6613e βˆ’ 16 6.1062e βˆ’ 15 βˆ’1.5282e βˆ’ 03 βˆ’2.5434e βˆ’ 04 βˆ’6.6614e βˆ’ 16 6.1062e βˆ’ 15

IT 243 32 156 144 243 19 156 144 243 18 156 144

𝑁𝑠 β€” β€” 96 85 β€” β€” 96 85 β€” β€” 96 86

Time 36.9898 4.1262 0.4067 0.3186 325.6083 25.1720 0.6852 0.5342 3218.5422 271.1651 5.1941 4.0120

IT 405 31 159 139 β€” β€” 159 139 β€” β€” 159 139

𝑁𝑠 β€” β€” 45 25 β€” β€” 45 25 β€” β€” 45 25

Time 1371.0198 95.1782 3.9951 2.9676 fail3 fail3 6.5173 4.6898 fail2 fail2 31.7107 25.1842

IT β€” 27 197 197

𝑁𝑠 β€” β€” 46 46

Time fail3 359.4507 20.8919 21.2641

(b)

π‘š 102

103

104

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

βˆ—

𝑓(π‘₯ ) 1.2524 0.1417 1.2510 1.2510 β€” β€” 1.2510 1.2510 β€” β€” 1.2510 1.2510

𝑔max (π‘₯βˆ— ) βˆ’1.0693e βˆ’ 02 βˆ’2.0927e βˆ’ 05 1.2490e βˆ’ 14 1.2490e βˆ’ 14 β€” β€” 1.2490e βˆ’ 14 1.2490e βˆ’ 14 β€” β€” 1.2490e βˆ’ 14 1.2490e βˆ’ 14 (c)

π‘š 102

Method ACH KNITRO FACH MFACH

𝑓(π‘₯βˆ— ) β€” 0.1099 1.1880 1.1880

𝑔max (π‘₯βˆ— ) β€” βˆ’5.6905e βˆ’ 05 2.3370e βˆ’ 14 2.4258e βˆ’ 14

12

Abstract and Applied Analysis (c) Continued.

π‘š

βˆ—

103

104

𝑔max (π‘₯βˆ— ) β€” β€” 2.3370e βˆ’ 14 2.4258e βˆ’ 14 β€” β€” 2.3370e βˆ’ 14 2.4258e βˆ’ 14

𝑓(π‘₯ ) β€” β€” 1.1880 1.1880 β€” β€” 1.1880 1.1880

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

IT β€” β€” 197 197 β€” β€” 197 197

𝑁𝑠 β€” β€” 46 46 β€” β€” 46 46

Time fail2 fail2 28.8674 27.5140 fail2 fail2 101.3199 99.1354

IT β€” β€” 260 258 β€” β€” 260 258 β€” β€” 260 258

𝑁𝑠 β€” β€” 47 46 β€” β€” 47 46 β€” β€” 47 46

Time fail3 fail3 126.6290 125.7973 fail2 fail2 146.6079 145.7293 fail2 fail2 368.5331 369.3701

IT 1605 2 815 820 1605 2 815 820 1606 2 960 931

𝑁𝑠 β€” β€” 288 293 β€” β€” 288 294 β€” β€” 6444 7094

(d)

π‘š 102

103

104

βˆ—

𝑔max (π‘₯βˆ— ) β€” β€” βˆ’6.2839e βˆ’ 14 4.0967e βˆ’ 14 β€” β€” βˆ’6.2839e βˆ’ 14 4.0967e βˆ’ 14 β€” β€” βˆ’6.2839e βˆ’ 14 4.0967e βˆ’ 14

𝑓(π‘₯ ) β€” β€” 1.1406 1.1406 β€” β€” 1.1406 1.1406 β€” β€” 1.1406 1.1406

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

Table 4: Test results for Example 18. π‘š

𝑓(π‘₯βˆ— ) 5.3347 1.2000e + 05 5.3347 5.3347 5.3347 1.2000e + 05 5.3347 5.3347 5.3347 1.2000e + 05 5.3347 5.3347

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

102

104

106

𝑔max (π‘₯βˆ— ) βˆ’4.4409e βˆ’ 16 βˆ’3.9900e + 02 1.3323e βˆ’ 15 1.3323e βˆ’ 15 βˆ’4.4409e βˆ’ 16 βˆ’3.9900e + 02 1.3323e βˆ’ 15 1.3323e βˆ’ 15 1.3323e βˆ’ 15 βˆ’3.9900e + 02 βˆ’9.2480e βˆ’ 08 βˆ’1.1278e βˆ’ 07

Example 19 (see [22]). Consider

Example 20 (see [22]). Consider

𝑓 (π‘₯) = (π‘₯1 βˆ’ 2π‘₯2 + 5π‘₯22 βˆ’ π‘₯23 βˆ’ 13) + (π‘₯1 βˆ’ 14π‘₯2 +

π‘₯22

+

π‘₯23

𝑓 (π‘₯) =

2

𝑖 , (π‘š βˆ’ 1)

π‘₯0 = (0, βˆ’45) ∈ 𝑅2 .

2

βˆ’ 29) ,

𝑖 = 0, . . . , π‘š βˆ’ 1,

π‘₯12 π‘₯1 + + π‘₯22 , 3 2

𝑔𝑖 (π‘₯) = (1 βˆ’ π‘₯12 𝑑𝑖2 ) βˆ’ π‘₯1 𝑑𝑖2 βˆ’ π‘₯22 + π‘₯2 ,

2

𝑔𝑖 (π‘₯) = π‘₯12 + 2π‘₯2 𝑑𝑖2 + exp (π‘₯1 + π‘₯2 ) βˆ’ exp (𝑑𝑖 ) , 𝑑𝑖 =

Time 0.7546 0.0141 0.2977 0.3004 7.0581 0.3163 1.3127 1.3185 575.4663 3243.2201 42.7664 41.7933

(60)

𝑖 𝑑𝑖 = , (π‘š βˆ’ 1)

(61)

𝑖 = 0, . . . , π‘š βˆ’ 1,

π‘₯0 = (βˆ’1, 100) ∈ 𝑅2 . To explain the numerical efficiency, we make the following remarks by numerical results in Tables 1, 2, 3, 4, 5 and 6.

Abstract and Applied Analysis

13 Table 5: Test results for Example 19.

π‘š 102

104

106

βˆ—

Method ACH KNITRO FACH MFACH ACH KNITRO FACH MFACH

𝑓(π‘₯ ) 97.1589 97.1592 97.1589 97.1589 97.1589 102.2095 97.1589 97.1589

𝑔max (π‘₯βˆ— ) 0.0000 βˆ’7.5266 βˆ’ 05 0.0000 0.0000 0.0000 βˆ’6.7465 βˆ’ 01 0.0000 0.0000

IT 213 68 170 160 227 263 249 251

𝑁𝑠 β€” β€” 191 183 β€” β€” 10926 11321

Time 0.0964 0.5162 0.0795 0.0608 0.5310 15.4868 0.2846 0.2783

ACH KNITRO FACH MFACH

97.1589 188.9496 97.1589 97.1589

0.0000 βˆ’8.3841e βˆ’ 01 0.0000 0.0000

229 17 334 253

β€” β€” 427520 486506

50.8763 2206.4660 9.5724 6.8910

Table 6: Test results for Example 20. Method

𝑓(π‘₯βˆ— )

𝑔max (π‘₯βˆ— )

IT

𝑁𝑠

Time

10

ACH KNITRO FACH MFACH

2.4305 2.4305 2.4305 2.4305

0.0000 βˆ’1.3822e βˆ’ 10 2.2204e βˆ’ 16 2.2204e βˆ’ 16

342 21 348 341

β€” β€” 387 374

0.1749 0.0291 0.1368 0.1276

104

ACH KNITRO FACH MFACH

2.4305 2.4305 2.4305 2.4305

βˆ’6.6632e βˆ’ 09 βˆ’1.6180e βˆ’ 10 βˆ’6.6632e βˆ’ 09 βˆ’6.6632e βˆ’ 09

344 38 296 358

β€” β€” 14710 23559

0.9804 1.0329 0.2610 0.2802

106

ACH KNITRO FACH MFACH

2.4305 2.6182 β€” 2.4305

βˆ’4.9783e βˆ’ 08 βˆ’1.2609e βˆ’ 04 β€” βˆ’4.9021e βˆ’ 08

344 61 β€” 516

β€” β€” β€” 2231308

94.3943 2372.2093 fail1 16.3756

π‘š 2

(i) If 𝑔max (π‘₯) ≀ βˆ’π›Όπœ€(𝑑) for any (π‘₯, πœ†, 𝑑) ∈ Γ𝑀0 , the FACH and MFACH methods do not need to calculate the gradient and Hessian of any constraint functions. (ii) For problems whose gradients and Hessians of constraint functions are expensive to evaluate, the performance of the FACH and MFACH methods is much better than the ACH method based on similar numerical tracing procedures. (iii) Compared to the interior-point direct algorithm of the state-of-the-art solver KNITRO, the test results show that the FACH and MFACH methods perform worse when π‘š is small but much better when π‘š is large for most problems. In addition, we can see that their time cost increases more slowly than KNITRO solver as π‘š increases. (iv) The function πœ‘(𝑧, 𝑑) is important for the flattened aggregate constraint function. Theoretically, the parameters 𝑐1 , 𝑐2 , and 𝛼 in the FACH and MFACH methods can be chosen freely. However, they do matter in the practical efficiency of the FACH and MFACH methods and should be suitably chosen. If

these parameters are too large, then too many gradients and Hessians of individual constraint functions need to be evaluated and, hence, cause low efficiency. On the other hand, if they are too small, the Hessian of the flattened aggregated constraint function (10) may become ill-conditioned. In our numerical tests, we fixed 𝑐1 = 0.05, 𝑐2 = 0.5 βˆ— 10βˆ’5 , and 𝛼 = 2. In addition, the function πœ‘(𝑧, 𝑑) can be defined in many ways, and preliminary numerical experiments show that the algorithms with different functions πœ‘(𝑧, 𝑑) have similar efficiencies. (v) In algorithm FACH-S-N, we gave only a simple implementation of the ACH, FACH, and MFACH methods. To improve implementation of the FACH and MFACH methods, a lot of work needs to be done on all processes of numerical path tracing, say, schemes of predictor and corrector, steplength updating, linear system solving, and end game. Other practical strategies in the literature for large-scale nonlinear programming problems (e.g., [9, 23–26]) are also very important for improving the efficiency.

14

Abstract and Applied Analysis

5. Conclusions By introducing a flattened aggregate constraint function, a flattened aggregate constraint homotopy method is proposed for nonlinear programming problems with few variables and many nonlinear constraints, and its global convergence is proven. By greatly reducing the computation of gradients and Hessians of constraint functions in each iteration, the proposed method is very competitive for nonlinear programming problems with a large number of complicated constraint functions.

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

[12]

[13]

[14]

[15]

[16]

Acknowledgment The research was supported by the National Natural Science Foundation of China (11171051, 91230103).

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Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014